Raymond Firth's book The Interpretation of Ritual, published in 1972, describes:

Open-handed salutation may vary from the casual wave of greeting to a passing acquaintance or the more regular flapping of an uplifted wrist in farewell, to the precise military salute.
pg. 28

That's only fifty years ago, well within living memory, and yet I find the description of the flapping uplifted wrist as strange (though I've been alive less than three decades).

The paper "Cultural Similarities and Differences in Emblematic Gestures" by David Matsumoto studied gestures from different cultures with recognition rates of 70% or higher. At a 98.2% recognition rate was the US salutation described as

Hello: One hand raised and open, palm facing away from body; fingers or hand fluttered sideways or remains stationary

​

And at 100%:

Goodbye: One hand raised and open, palm facing away from body; fingers or hands fluttered back and forth or sideways, or hand remains stationary

Implying that goodbyes are allowed to flap forward; hellos are not.

Seeing this, I would speculate that perhaps the two gestures, being already so similar, have begun to converge, the more widely applicable hello-sideways format becoming the dominate format. But that's just my speculation... let's look at the history of the gesture.

According to "From Hat Honour to the Handshake: Changing Styles of Communication in the Eighteenth Century" by P.J. Corfield, informality and the military's need for efficiency evolved older traditions into what we now know as the salute:

• Originally, "Hats were to be removed with something of a flourish, when greeting men who were their equals and, especially, when encountering their social betters"
• Towards the late eighteenth century, "Gestures with headgear simultaneously became less extravagant. Gentlemen began to tip their hats lightly, rather than remove them entirely with a huge flourish. And lower-class men giving a mark of respect to their social « superiors » would give a slight pull to their caps – or, if hatless, tug their forelocks – or, if hatless and hairless (or simply careless), simply raise a hand to the head."
• Thus the motion of hand to head became symbolic of recognizing authority. "There were many stylistic variations... Over time, however, military forces gradually standardised their codes, to reduce confusion."

How does this relate back to hand waves? The explanation lies within a single line from Wikipedia: "The original gesture of waving was saluting." Unfortunately... this line has no citation.

So to summarize: I failed to pinpoint the exact when, though we can say that the goodbye-forward format was likely the dominant format back in 1972. When it comes to the why, however, the comparison to the salute gesture at least gives us greater understanding to how gestures can change over time in response to changing expectations of formality, or simply change for convenience (if an extravagant hat flourish can one day become a brisk raise of hand to head, a goodbye gesture converging into a similar hello gesture seems like a reasonable hypothesis to me).

Side note: If anyone wants to check the citations on Wikipedia's page for "Wave (gesture)", you'll need the internet archive for this one.

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